so the background that come with the exo terra terrarium...about three inches of this will be sitting in water all the time...
will this start to rot and fall apart after time
Agreed - rot is not a concern. But it will absorb plenty of water. And leach dyes and gases into the air and water. Leaching is accelerated with the water absorption. Yum.
I personally wouldn't have it sitting in water. Why not just hack off the bottom, so it comes down to just above the substrate level? That way it's less prone to the negative issues mentioned above. (I assume you're going to adhere it to the back of the tank, so it can "levitate".)
Cutting it will
unleash a mess of "balls" with super-annoying static cling. Just cover the cut edge (or better yet, the whole damn thing!) with something pet-safe and encapsulating. E.g. acrylic modeling paint. Or epoxy, such as Pond Shield. The latter comes in various colors and also can be painted. I'm confident you can do a better paint job than Exo provides. You did model railroading? Yeah, you've probably got some art skills.
Finally - are you sure
you don't want to just drill & drain? Skip the whole tube, cap, basting routine? For real, it is nothing whatsoever
to be intimidated by.
In closing - this may all feel like an unwelcome speed bump. However the advice is borne of experience. In the long haul IMO you'd get better performance and thus satisfaction this way, than how you're setting out. Free advice though - take it, modify it, reject it outright, as you will.
Have fun, and good luck!